Magnetic Effects: Biot-Savart & Ampere's Law

The magnetic field is produced by moving charges (currents). Unlike the electric field, magnetic field lines always form closed loops (no magnetic monopoles). The Biot-Savart law gives the field due to a small current element: $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l} \times \hat{r}}{r^2}$, where $\mu_0 = 4\pi \times 10^{-7}$ T-m/A is the permeability of free space.

Biot-Savart Law — Current Element Diagram:

Magnetic Field Due to Standard Configurations:

Magnetic Field Around a Straight Current-Carrying Wire:

Straight wire: At perpendicular distance $d$ from an infinitely long wire carrying current $I$: $B = \mu_0 I/(2\pi d)$.
For a finite wire subtending angles $\theta_1$ and $\theta_2$ at the point: $B = \frac{\mu_0 I}{4\pi d}(\sin\theta_1 + \sin\theta_2)$. Direction: concentric circles around wire (right-hand thumb rule).

Circular loop: At the center of a loop of radius $R$: $B = \mu_0 I/(2R)$.
On the axis at distance $x$ from center: $B = \mu_0 IR^2/(2(R^2+x^2)^{3/2})$.
For $x \gg R$: $B \approx \mu_0 IR^2/(2x^3) = \mu_0 m/(2\pi x^3)$ where $m = I\pi R^2$ is the magnetic moment.

Force on a Current-Carrying Conductor in a Magnetic Field:

Force on a Current-Carrying Conductor: $\vec{F} = I\vec{L} \times \vec{B}$ (for straight conductor in uniform field). Force per unit length between two parallel wires: $F/l = \mu_0 I_1 I_2/(2\pi d)$. Parallel currents attract; antiparallel repel.

Ampere's Circuital Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$. The line integral of $\vec{B}$ around any closed loop equals $\mu_0$ times the net current enclosed. Most useful for symmetric current distributions.

Solenoid — Uniform Field Inside:

Solenoid: Inside an ideal solenoid of $n$ turns per unit length: $B = \mu_0 nI$ (uniform).
Outside: $B = 0$.
Toroid: Inside the toroid: $B = \mu_0 NI/(2\pi r)$ where $N$ is total turns and $r$ is the distance from the center.
Outside: $B = 0$.

Moving Coil Galvanometer: Torque on a current loop in a magnetic field: $\tau = NIAB$ (for $N$ turns, area $A$). The radial magnetic field (using concentric pole pieces) ensures $\tau \propto I$ for all deflection angles.
Current sensitivity: $I_g = NAB/k$ (deflection per unit current), where $k$ is the torsional constant.

The key problem-solving concept is choosing the right law: Biot-Savart for specific geometries (loops, finite wires), Ampere's law for highly symmetric distributions (infinite wires, solenoids, toroids).